8. In the ring Z8[x], show that [1] + [2]x is a unit.
9. Let R be a commutative ring with 1 and f(x) = a0 + a1x + ... + anxn ∈ R[x]. If f(x) is a unit in R[x], prove that a0 is a unit in R and ai is nilpotent for all i = 1, 2, ..., n.
10. Use the result of Exercise 9 to show that 1 + 5x is not a unit in Z[x].
11. Find all units of Z[x].
12. Find all units of Z6[x].
13. Let R be an integral domain. Prove that the units of R[x] are contained in R.
14. In Z8[x], prove the following.
(i) [4]x2 + [2]x + [4] is a zero divisor.
(ii) [2]x is nilpotent.
(iii) [4]x + [1] and [4]x + [3] are units.
15. Let R be a subring of a commutative ring S such that R has an identity.
(i) In the polynomial ring R[x1, x2, ..., xn], prove that x1, x2, ..., xn are algebraically independent over R.
(ii) Prove that the mapping ̑ : R[x1, x2, ..., xn] → R[c1, c2, ..., cn] defined by ̑( ∑in...i1 ri1...in xi11 ... xinn ) = ∑in...i1 ri1...in ci11 ... cinn is a homomorphism of R[x1, ..., xn] onto R[c1, ..., cn], where c1, ..., cn ∈ S.
(iii) Prove that the homomorphism ̑ in (ii) is an isomorphism if and only if c1, c2, ..., cn are algebraically independent over R.
16. Let f(x) be a polynomial of degree n > 0 in a polynomial ring K[x] over a field K. Prove that any element of the quotient ring K[x] / (f(x)) is of the form g(x) + (f(x)), where g(x) is a polynomial of degree at most n - 1.
17. For the following statements, write the proof if the statement is true; otherwise, give a counterexample.
(i) If a polynomial ring R[x] has zero divisors, so does R.
(ii) If R is a field, then R[x] is a field.
(iii) In Z7[x], (x + [1])7 = x7 + [1].